Practical audio power amplifiers using Pulse Width Modulation (PWM) have been known since the mid 1960s. In amplifiers from that era, a pulse train was generated by comparing a voltage representing the incoming audio signal with a reference waveform, typically a triangular wave or sawtooth wave, with a frequency in the range 50 kHz-200 kHz. The comparison yielded a 2-level rectangular wave having the same frequency as the reference waveform, and with a mark:space ratio varying in sympathy with the audio. The rectangular wave was amplified to the desired power level and then passively lowpass filtered to remove most of the high-frequency components of the rectangular wave, leaving its average level, which follows the audio, to drive a load such as a loudspeaker.
It is possible to obtain extremely good performance when such amplifiers are run ‘open-loop’, that is without feedback, but it is an expensive solution since the amplifier's performance is critically dependent on the quality of the output stages and the power supply. To alleviate these dependencies, the trend in the 1970s and subsequently has been to incorporate feedback. One simple way to incorporate feedback in an amplifier that compares the audio with a triangle wave, is to replace a fixed triangle wave by a sawtooth wave that is obtained by integrating the substantially rectangular wave that appears at the output of the amplifier's power switches. Analysis shows that this is an effective means of providing feedback. Moreover since the feedback is tightly integrated into the PWM itself, stability problems typically associated with feedback do not arise.
Amplifiers as described above have sometimes been called ‘digital’ in the popular press, but we shall describe them as ‘analog’, because the timings of the edges of the rectangular waves can vary continuously in sympathy with the audio. We shall reserve the word ‘digital’ for an amplifier in which the edge timings are quantized, so that the edge timings can be represented digitally and the edges can be generated by counting pulses produced by a high-precision, high-frequency clock, such as a crystal oscillator.
This principle was proposed by Sandler [6], who also realized that the apparent need for a clock frequency in the gigahertz region could be avoided by the use of oversampling and noise shaping. Several commercial products are now available that use this principle (see, for example, [3].)
The digital principle brings precision to the generation of the PWM waveform, but the power amplification, typically accomplished by MOSFET (Metal Oxide Silicon Field Effect Transistors) power switches, remains a fundamentally analog process, and as such is vulnerable to non-ideal component behavior. There is a distortion associated with the switching called “dead-time distortion”, and there is dependency on the power supply just as with the original analog PWM amplifiers. Without feedback or other compensation, the gain of the output stage will be directly proportional to the supply voltage. This precludes the use of an inexpensive non-regulated power supply, or condemns the system to relatively poor performance.
Attempts have been made in the prior art to apply feedback to the output stages of a digital PWM amplifier. One such attempt is embodied in the PEDEC (PCT/DK98/00133) principle, in which a modulator operating at a relatively low level produces a PWM waveform, and a correction unit re-times the edges of the waveform before passing the waveform to the power switches. The correction unit receives control signals from an error processing unit, which compares the original low-level PWM waveform with the output of the power switches. The input to the power switches is thus modified in dependence on the output, creating a feedback loop.
The PEDEC principle can be applied to a digital or an analog PWM amplifier. However the feedback is analog and local to the output stages—the quality of the output is fundamentally determined by the quality (including jitter properties) of the low-level PWM waveform.
Another example of feedback in the prior art is the disclosure by Melanson in U.S. Pat. No. 6,373,334 “Real Time Correction of a Digital PWM Amplifier”. Again, the feedback is derived by comparing a low-level square wave with the output of the power switches. In this proposal, however, the correction is fed back to the PWM modulator, so there do not exist two PWM waveforms, original and re-timed, as in the PEDEC proposal. U.S. Pat. No. 6,373,334 describes a feedback that is tightly integrated into a particular type of PWM modulator. It shares with PEDEC the property that the quality of the final output is limited by the quality of the low level PWM waveform.
In an analog (non-PWM) amplifier, it is customary to take at least some feedback from the final output to a point close to the input. A substantial reason why this is difficult in a digital PWM amplifier is loop delay. In particular, since the output is analog but the input and early processing are digital, an ADC (Analog to Digital Converter) is required in the feedback path. Depending on the topology, the quality of the final output will be directly related to the quality of the ADC. Currently available audio ADCs of sufficient quality, however, have delays that are completely excessive for inclusion in a loop that provides significant feedback over the audio range of 0-20 kHz.
Even when the ADC delay has been minimized, substantial stability problems remain. There is an extensive literature on stabilizing feedback loops, using Bode plots, lead/lag compensation, nested feedback and the like. Most of the techniques apply to linear systems with constant gain, and there is little guidance on how to deal with nonlinearity or gain variation apart from allowing an adequate “gain margin” or “phase margin”.
Unfortunately, a loop that includes a delay of, for example, 10 μs, and that has enough “gain margin” or “phase margin” to be robust against nonlinearity and gain variation, is unlikely to provide a significant degree of feedback at 20 kHz. “Nested feedback” appears at first sight to be able to provide large amounts of feedback with stability. On examination, however, it is found that the stability is “conditional”, which means that it is susceptible to gain variation, and oscillation can be caused even by a reduction of the gain of the forward path. Consequently, this technique would be completely unsuitable for use in a PWM amplifier that is required to work with an unregulated power supply.
A less obvious problem is the intrinsic nonlinearity introduced by the pulse width modulation process. This is normally thought of as a small effect that introduces harmonic distortion at high audio frequencies (e.g., −70 dB 3rd harmonic on a full scale 5 kHz fundamental [3].) However, design of a feedback loop requires one to consider frequencies well outside the band that is effectively controlled by feedback. In the case of a digital PWM amplifier with a sampling and switching frequency of 384 kHz, frequencies up to the Nyquist of 192 kHz should ideally be considered. At 192 kHz, the forward gain of a conventional double-edge PWM modulates by 100% as the mark:space ratio of the PWM waveform varies over its full range. Even at 80 kHz, the forward gain modulates by 20%. Such modulation of a part of the spectrum that is only two octaves above the top of the range that is desired to be controlled will set a limit to how “aggressive” any conditionally stable feedback can be, even for amplifiers that are always used with stabilized power supplies.
Several correction methods are known for PWM nonlinearity. One straightforward method, as shown in [3], achieves almost complete cancellation of the nonlinear effect within the audio band. However if it is hoped that feedback stability will be improved by correcting the PWM nonlinearity, then the corrector must be placed inside the feedback loop. Since the corrector in [3] has a delay of one sample (e.g. 2.6 μs) the stability problem is already worse. Further, while the correction is almost perfect within the audio band, it still does not provide consistent performance near the Nyquist frequency, for it is not possible to compensate a gain modulation of 100%.
In view of the difficulties discussed above, there is a need for a robust method for applying feedback to a digital PWM amplifier that directly addresses the issues of loop delay, nonlinearity and variation in the forward gain.